Effect of nonlinearity on Head-Tail instability

3/18/04

contents

• 2-particle model and head-tail instability

• Nonlinear force– One particle under nonlinear force– Decomposition of the results– Nonlinear resonance under sinusoidal driving– Is this a way to simplify the dynamics of 2-

particle model under nonlinear force

• Conclusion

2-particle model

• A bunch made of two macro-particles, each of them executes synchrotron oscillation.

• The tail-particle sees the wake field produced by head-particle.

• After each half synchrotron period, the two particles switch longitudinal positions

Equations of motion:

122

2

12

1 0

xFxx

xx

w

• Where, is the wake force, and is related to by: *

• is a dimensionless parameter • Solutions stable condition: , when

wF

s

wF2

2

* See A. Chao “physics of collective beam instabilities in high energy accelerator”

s

Modes of the model

22sin

odd ,2

even ,2

ll

ll

ss

ss

tune

Sextupole and damping

122222

22

21211

21

2

0 2

xFxgxxx

xgxxx

w

Can be unstable when is smaller than 2

s

wF2

Head particle motion in phase space

FFT of equation-1 solution

Tune shift vs. Xmax

Ratio of amplitudes vs. Xmax

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.0 0.5 1.0 1.5 2.0 2.5

Xmax (mm)

Rat

io o

f am

pli

tud

esg2=10000 g2=20000

Ratio of amplitudes vs. tune shift

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

tune shift

Rat

io o

f am

plitu

de

g2=10000 g2=20000

Sinusoidal driving nonlinear oscillation

)cos('2'' 22

2 daxgxxx

Fold over effect

Driving frequency

Xm

ax

Transition point

Xm

ax

X_initial

Bi-stability

In phase space

In phase space with larger driving

In phase space with larger driving

Compare the approximation with the total wake case

Total w

ake

Fundam

ental com

ponent only

2-particle model

Conclusion

• The wake force from head particle is approximated with a simple harmonic solution.

• The equation of motion of tail particle becomes a sinusoidal driving nonlinear oscillation.

• Based on understanding of motion of the single particles, further work instability for 2-particle model under nonlinear force can be better investigated.